Option pricing with perturbation methods

Abstract

This thesis discusses the use of perturbation theory in the context of financial mathematics, in particular on the use of matched asymptotic expansions in option pricing. Our methods are applied to the ordinary Black-Scholes model for illustration. In this simple example of the Black-Scholes model an exact solution is available, so it is in fact not neccessary to apply the method of asymptotic expansions on this model. However, in case we do apply the method, two artificial layers have to be constructed. Making smart choices for the local variables leads to a transformation of the equations into a heat equation, which can easily be solved. Finally, the results are compared to a Taylor expansion of the exact solution to see that this method is very accurate. After this first instructive model, the method of matched asymptotic expansions is applied to two more advanced models based on papers by Sam Howison and Patrick Hagan et al.. Here, different choices for the scalings are made. The former discusses a fast mean-reverting stochastic volatility model that turns out to have many open ends. In Howison's paper quite a lot of assumptions and simplifications are made. Unfortunately, often the motivation for them is not explicitly given in the paper, and in some cases we even think these assumptions and simplifications are incorrect. The latter examines a new three-parameter stochastic volatility model that successfully prices back the volatility smile as observed in the market nowadays, and that is commonly used. The derivation of this model is the main focus of this thesis. The resulting expression for the implied volatility under the SABR model is obtained by considering the forward and backward Kol-mogorov equations per order in epsilon, making some smart choices for local variables and functions in order to transform them into an equation that looks like a heat equation, which is easier to solve. Recommendations for further investigation on these models would be to consider several different choices for the scalings and see which one works best.